How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: MoeBlee on 3 Jul 2010 16:44 On Jul 3, 4:53 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > in all references given (Gentzen by Frederick Williams and > Hinman by MoeBlee) there is no mention of any of ZFCs axioms anywhere > in their proofs (including earlier results on which it relies) at > all. Hinman's proof concerns ZF (so ZFC a fortiori). Of course certain of the ZF axioms are used in Hinman's proof as that proof depends on certain results he references from earlier chapters. MoeBlee
From: Jack Campin - bogus address on 3 Jul 2010 17:20 >> What Charlie-Boo needs, therefore, is some evidence that >> convinces him that if one were to work out all the steps, one >> can eventually prove Con(PA) in ZFC, just as one can eventually >> find googolduplex digits of sqrt(2)+sqrt(3). So far, no poster or >> book has so convinced him. Therefore, there is no reason for >> him to believe that such a proof exists. > Of course. But it's just a mathematical theorem that virtually anyone > who is informed in the basics of the subject can do for himself. No > one can give a proof for Charlie-Boo that will convince him, since a > proof of this theorem depends on lots of terminology, formulations, > and previously proven theorems in Z set theory, which he refuses to > learn (let alone the predicate calculus). You don't expect someone to > prove in a post or two some result in mathematics such as analysis, > abstract algebra, etc. that requires first learning the basics of > those subjects, do you? Same for mathematical logic. I was wondering what position C-B would take on the model of Euclidean geometry provided by tuples of real numbers. The model dates back to Descartes (details filled in by various people up to Hilbert) and it's a bit more complicated to verify than PA in ZFC. But despite being a similar argument, it's in some sense more "ordinary" mathematics, so it doesn't get the cranks going. Or does it? Does C-B think algebraic geometry is illegitimate? ----------------------------------------------------------------------------- e m a i l : j a c k @ c a m p i n . m e . u k Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland mobile 07800 739 557 <http://www.campin.me.uk> Twitter: JackCampin
From: MoeBlee on 3 Jul 2010 17:26 On Jul 3, 1:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 1, 1:40 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> MoeBlee wrote: > >> A crank would "say" anything too! But I've never believed Aatu is a crank > >> so where's his _proof_, in FOL level or meta level? Oh, but you're going > >> to explain the "proof" right below, I see. > > > WHAT proof? Proof of WHAT? Proof of the consistency of PA? There are > > many. > > So there are many (formal) proofs of the consistency of PA, after all! OF COURSE! I've said that all along. And I've said that one is quite reasonable to view that such proofs have no epistemological value. This is all discussed beautifully in layman's terms in Franzen's book. OF COURSE if one doubts the consistency of PA, then a proof, from an even STRONGER theory, such as Z, provides no basis for alleviating said doubts. When I say there is a proof in this contexgt, I mean 'proof' in the technical sense of a formal proof, a derivation using recursive rules of inference with a recursive set of axioms, not necessarily in the sense of "indisputably convincing basis for belief" or related such senses. Such a thing may well not provide adequate basis for BELIEF if one does not already have adequate basis to believe said axioms are true. > > But Aatu is saying such proofs are NOT the basis for his > > conviction that PA is consistent. > > So are you saying Aatu is a crank, because he didn't base his assertion > that PA is consistent on proofs, but only on his "conviction", as you > claimed below? No, because my definition of the informal "sociological" rubric 'crank' (I've given it in other posts; I'm not going to type it up again) does not entail that one is a crank merely for having pre- formal mathematical beliefs. > >>> His basis is for that is not a FORMAL proof, but rather his > >>> conviction that the axioms of PA are true (and not even in confined to > >>> a FORMAL model theoretic sense of truth, but rather that the axioms > >>> are simply true about the natural numbers, as we (editorial 'we') > >>> understand the natural numbers even aside from any formalization. > >> Let me see: his proof > > >> - isn't based on rules of inference and axioms > >> - isn't based on "model theoretic sense of truth" > >> - is merely based on _conviction_ that "the axioms of PA are > >> true" and our intuitive knowledge of the natural numbers > >> "aside from any formalization". > > > I didn't say it is a PROOF. Why are you not LISTENING? > > Ah I see! So to you, Aatu just said things without proof here? Aatu and lots of people say things without proof. So what? Without proof I say that the lollipop I'm eating now is orange flavored. So what? Without proof, I say that I believe every theorem of PRA is true in a finitistic sense of 'true'. So what? > > I didn't post ANY definition of the consistency of a formal system. I > > posted a definition of consistency of a set of formulas. > > So, is the 2-formula set I gave a consistent set of formulas on your > definition? (The first one that you emphatically ended with "PERIOD.") I revised my definition (as I realized I had misstated my actual definition in my notes). If you want to know whether a certain set is consistent under that definition, refer to my revised definition. MoeBlee
From: MoeBlee on 3 Jul 2010 17:35 On Jul 3, 2:04 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > > I didn't state that as MY view. I was telling you AATU's view. For > > Christ sakes! > > I was asking Aatu, _not you_ to clarify about relative consistency > and you "represented" him, even though he had never asked you to do > so! I don't presume to represent him in general. Only as to those specific remarks, I chose to state what his view is. At worst, I'm guilty of being presumptuous to state what his view is. But I have sufficiently discussed the subject of consistency of PA with Aatu and read enough of his posts to have a clear understanding of his view to the extent I stated it. > So you did represented Aatu in his absence! Therefore in this context > I can't make a distinction between you or him. Then there's no basis for rational discussion with you. I said that I was stating what Aatu's view is but that it is not necessarily my view (the ways in which I depart from Aatu are somewhat philosophically complicated and our differences are found in certain posts we made to each other; as well as some further considerations I would have to explain). By saying "This is what Aatu is saying, ..." doesn't entail that my own views are the same as his. I can't imagine a reasonable person not allowing that. > If you don't want it > that way, then don't represent anybody! Just argue on your own. I'll do as I please. I chose to state that Aatu's view is such and such in order to make that matter stark for you, as pleased me to do. If you claim that obligates me to be regarded as having the same view as Aatu then that's just more of your mulishness. > As far as I'm concerned, you just hide behind Aatu's name to "blast" > me (and you did here a few times) without using your own credibility! You really are a nut. MoeBlee
From: MoeBlee on 3 Jul 2010 17:54
On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > >> So there are many (formal) proofs of the consistency of PA, after all! > > > OF COURSE! I've said that all along. And I've said that one is quite > > reasonable to view that such proofs have no epistemological value. > > This is all discussed beautifully in layman's terms in Franzen's book. > > OF COURSE if one doubts the consistency of PA, then a proof, from an > > even STRONGER theory, such as Z, provides no basis for alleviating > > said doubts. > > > When I say there is a proof in this contexgt, I mean 'proof' in the > > technical sense of a formal proof, a derivation using recursive rules > > of inference with a recursive set of axioms, not necessarily in the > > sense of "indisputably convincing basis for belief" or related such > > senses. Such a thing may well not provide adequate basis for BELIEF if > > one does not already have adequate basis to believe said axioms are > > true. > > Let me try to summarize what you said above. There's no syntactical > proof that can possibly confirm the fact that PA is consistent, > if PA is in fact consistent. If this is what you meant I'm OK with > that. By 'proof' in this context I mean formal proof. I don't know what you find bundled with the word 'confirm', so I prefer to stand by what I posted, which is clear enough, especially as it is an extremely common notion discussed widely. > And if that's the case, would you be able to make a stand and say we > in fact just don't know PA if is consistent? I don't view the matter in that simple way of putting it even. > That would help to save a lot of ridiculous arguments. You'd do best not to argue as if I hold certain views I've never stated holding. A good many of my views are posted over the years. Moreover, on those matters in which I have not filled in my views, then, to be reasonable, you can only conclude that I have not yet had time, inclination, or achieved a conclusion to post on the matter. MoeBlee |